A126152 Main diagonal of symmetric triangle A126150: a(n) = A126150(n,n).

1, 4, 36, 744, 28536, 1736064, 152914176, 18372559104, 2885671339776, 573765893121024, 140835811776316416, 41820352964911908864, 14774712204104658671616, 6124078747943873540112384

Offset: 0

Author

Paul D. Hanna, Dec 19 2006

Keywords

• nonn

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 0..232

Crossrefs

Cf. A126150; A126151 (column 0), A126153 (diagonal).

Programs

PARI

/* Continued fraction involving even-indexed pentagonal numbers: */
{a(n)=local(CF=1+x*O(x),m,P); for(k=1, n,m=2*((n-k)\2+1);P=m*(3*m-1)/2-((n-k+1)%2); CF=1/(1-P*x*CF)); polcoeff(CF, n, x)}
for(n=0,20,print1(a(n),","))

Formula

a(n) = Sum_{k=0..n} A130847(n,k)*3^k. - Philippe Deléham, Jul 22 2007

G.f.: 1/(1 - 4*x/(1 - 5*x/(1 - 21*x/(1 - 22*x/(1 - 50*x/(1 - 51*x/(1 - 91*x/(1 - 92*x/(1 -...))))))))), a continued fraction involving even-indexed pentagonal numbers A000326. - Paul D. Hanna, Feb 15 2012

a(n) ~ Gamma(1/3) * 2^(3*n+7/3) * 3^(n+3/2) * n^(2*n+7/6) / (exp(2*n) * Pi^(2*n+13/6)). - Vaclav Kotesovec, May 30 2015